The abundance and persistence of Caprinae populations

Stable or growing populations may go extinct when their sizes cannot withstand large swings in temporal variation and stochastic forces. Hence, the minimum abundance threshold defining when populations can persist without human intervention forms a key conservation parameter. We identify this threshold for many populations of Caprinae, typically threatened species lacking demographic data. Doing so helps triage conservation and management actions for threatened or harvested populations. Methodologically, we used population projection matrices and simulations, with starting abundance, recruitment, and adult female survival predicting future abundance, growth rate (λ), and population trend. We incorporated mean demographic rates representative of Caprinae populations and corresponding variances from desert bighorn sheep (Ovis canadensis nelsoni), as a proxy for Caprinae sharing similar life histories. We found a population’s minimum abundance resulting in ≤ 0.01 chance of quasi-extinction (QE; population ≤ 5 adult females) in 10 years and ≤ 0.10 QE in 30 years as 50 adult females, or 70 were translocation (removals) pursued. Discovering the threshold required 3 demographic parameters. We show, however, that monitoring populations’ relationships to this threshold requires only abundance and recruitment data. This applied approach avoids the logistical and cost hurdles in measuring female survival, making assays of population persistence more practical.

The parameter values and combinations we employ correspond to empirical values reported from many different Caprinae populations. Indeed, estimating the minimum viable population (MVP) by evaluating a spectrum of biologically plausible values covers the differences in Caprinae social systems and environments influencing a given populations survival and recruitment values. We never assume that values for all Caprinae are similar, nor do we consider data from desert bighorn sheep representative for all Caprinae. We included model code and all simulation output, so any user can select demographic values of choice and examine the resulting changes in population persistence, or tailor an investigation to a specific population of interest.
Despite the copious literature defining minimum population sizes and building demographic simulations for a variety of species, prior studies rarely consider the practicality that managers require to inform on-the-ground work 12 . Such answers hinge on using data that managers can logistically acquire, affordably. Herein we implement this applied and practical approach.
We quantify and report the relationships between recruitment and adult female survival with population growth (λ), and therefore, the combinations of recruitment and survivorship required to attain a given λ, as recruitment and survival have most influence on Caprinae abundance 13,14 . Recruitment represents the number of offspring produced by reproductive females, each year, entering the yearling age class. Acquiring data describing recruitment is inexpensive and logistically simple. In contrast, survival data are usually derived from catching, marking (i.e., telemetry collars) and following adult females. The work is logistically challenging and expensive, making data describing survivorship scarce. It is also unlikely that biologists will acquire survivorship data for many Caprinae anytime soon. Therefore, while our simulations incorporate survivorship data, monitoring for the threshold abundance does not. Instead, we simplify results so biologists interested in determining when a population is secure from localized extinction can rely solely on accurate measures of abundance, abundance trend and recruitment.
Our results identify which demographic parameter generates the greatest impacts for increasing growth rate (λ) to recover an ailing population. We also show effects of translocation (adult female removals) on population abundance and trend of the source population, to identify populations sizes at which translocation is not detrimental. Lastly, we assess if survival or recruitment are useful for evaluating the efficacy of management actions, as the variability inherent to these parameters challenges such assessments.
For Caprinae managers, our results likely represent the sole information describing population projections and minimum abundance thresholds for the endangered Caprinae populations under their charge. For other wildlife and conservation practitioners, we exemplify a procedure for modeling population trajectories and identifying demographic thresholds for species with limited data.

Results
Demographic parameters. Wildlife agencies often produce lamb:adult female ratios (i.e. recruitment) from surveys. To quantify variability in recruitment, we analyzed recruitment data of desert bighorn sheep from three state agencies (USA): the Arizona Game and Fish Department (AZGFD), the California Department of Fish and Wildlife (CAFW), and the New Mexico Department of Game and Fish (NMDGF). The yearlings of Caprinae, of either sex, can be challenging to identify as their size and horn characteristics often mimic adult females. Agencies handle this situation differently. The CAFW and NMDGF report the number of yearling females identified but omit yearling males and unclassified yearlings. The AZGFD reports the number of male, female and unclassified yearlings. The NMDGF includes yearling females in the denominator of the lamb:ewe ratio but CAFW and AZGFD do not. Across the three states, the lamb:ewe ratios that omit yearlings from the denominator (L:E) and those that include yearlings in the denominator (L:EY) are generally similar (Table 1).
When yearlings counts are removed from the denominator of the L:E ratio, an unknown, incomplete and inconsistent fraction of yearlings are actually removed from each survey. For example, 46% of surveys for Table 1. Recruitment data from desert bighorn sheep populations during years with adult female (ewe) abundance ≥ 20. The ratios labeled L:E represent the mean lamb:ewe ratio with yearling counts excluded from the denominator, and L:EY represents the mean lamb:ewe ratio with yearling counts included in the denominator (90% CIs; LCL-Lower, UCL-Upper, SD indicates standard deviation). The column "n" reports the number of site-years included in the calculation. Acronyms are as follows: AZGFD Arizona Game and Fish Department, CAFW California Department of Fish and Wildlife, NMDGF New Mexico Department of Game and Fish. We estimated mean recruitment (L:EY) as 0.294 with a biological variance of 0.028 (n = 757; CV = 0.57). The mean and variance using the traditional L:E ratio (i.e., assumed exclusion of all yearlings from the denominator) was 0.33 and 0.033, respectively. Our subsequent analyses always used the biological variance with yearlings included in the ratio denominator.
The mean recruitment we employ aligns with values reported from other populations of Caprinae, and the variability in recruitment matches the reduced versions of variability we tested. For instance, populations of Dall's sheep (O. dalli dalli) have lamb:(ewe + yearling) ratios ~ 0.30 with CVs exceeding 0.4 15 . Mountain goats (Oreamnos americanus) have mean kid:(yearling + nanny) ratios of 0.33 and CV = 0.21 (calculated from Table 1 found in 16 ). Punjab urial (Ovis vignei punjabiensis) can have mean lamb:ewe ratios of 0.44 with CV = 0.18 17 , and Astor markhor (Capra falconeri falconeri) have mean kid:(yearling + nanny) ratios of 0.32 and CV = 0.30 (number of females ≥ 20; N = 8 18 ). The variability in our recruitment data is likely higher than reported elsewhere for many other Caprinae populations, because our method accounts for inter-annual variability in the calculation (not calculating the mean variance).
We estimated the mean survival for adult female desert bighorn sheep as 0.82, with biological variance = 0.0127 (CV = 0.14; 31 site/years). This mean and variance in survivorship also corresponds with other Simulations and minimum abundance of adult females. We quantified the probabilities of λ < 1 and quasi-extinction at year 10 and 30 (QE(10); QE(30)), with different combinations of mean adult female survival, mean recruitment, and starting abundance of adult females, with the variance values fixed (Fig. 1). To achieve a stable population (λ ~ 1.0), the lowest recruitment can become is 0.20, provided an adult female survival of 0.98 (deterministic λ = 1.01; Table 2; Fig. 1). Alternatively, recruitment could be as high as 0.45 with a corresponding survivorship of 0.75 (deterministic λ = 1.01; Table 2; Fig. 1).
The variability surrounding λ is high, with confidence levels often spanning ± 30% ( Table 2). The likelihood of QE helps represent this variability. Acquiring We also examined variability by quantifying the proportion of scenario iterations resulting in λ < 1 (Fig. 1, Table 2). Starting abundance has relatively small effects on this probability (Fig. 1). Attaining a ≤ 25% chance of population decline (stochastic λ) requires combinations of recruitment and survivorship such as survivorship ≥ 0.95 and recruitment 0.30, survivorship ≥ 0.90 and recruitment 0.40, or survivorship ≥ 0.85 and recruitment 0.50 ( Fig. 1). Attaining a ≤ 10% chance of population decline requires survivorship ≥ 0.95 and recruitment ≥ 0.50 ( Fig. 1).
We determined the minimum abundance of adult females required for ensuring population persistence, when the population has demographic rates consistent with a stable to slightly growing population (stochastic λ ~ 1; Supplementary Data S1). Results focused on QE, because starting abundance strongly affects this probability.  (Table 2). With a starting abundance of 100 adult females, a population has QE(10) < 0.01 and QE(30) ≤ 0.10 for all simulations. For all starting abundances, the risk of QE increases with high recruitment and low survival (and vice-versa; Table 2). Based on these results, a population of 50 adult females forms a reasonable target to ensure population persistence. Therefore, a population of 50 adult females remains a minimum threshold target, provided that frequent and accurate monitoring of abundance catches any decline.
Biological variation influences the potential trajectories of population growth or decline. Higher variability means more alternative paths, so reductions in variability typically provide greater assurances in a population's projection. We found that arbitrarily reducing biological variation by half for recruitment or adult female survival barely changed the resulting variability surrounding λ (Table 4). Nor does this reduction in variability substantially influence a population's trajectory (Fig. 3). Keeping biological variance for one parameter and eliminating variance for the other, or halving biological variation for both parameters, reduces variability surrounding a stable λ to 20%, with little effect on population trajectories (Table 4; Fig. 3). Reducing variability by 90% or eliminating it for both parameters generated confidence intervals spanning ~ 10% around a stable λ ( www.nature.com/scientificreports/ 10 ( Fig. 3). Increases in biological variation raise QE, but effects do not manifest at year 10 (QE(10) always 0), and QE (30) changes are minimal ( Table 4).

Examination of demographic parameters.
We determined the effects of temporal and geographical (across population) variability. Given 9-10 consecutive surveys, mean recruitment and biological variance were 0.30 and 0.028 (n = 11; CV = 0.56), matching results from all these recruitment data. Within a 5 year consecutive period, recruitment = 0.31 with variance = 0.021 (n = 24; CV = 0.48). For 3 year consecutive periods, the mean recruitment ratio = 0.29 with variance = 0.018 (n = 73; CV = 0.46). Reducing data duration lowered the recruitment variance. Mean values remained consistent. We examined geographical variation in recruitment data (L:EY) with populations ≥ 20 adult females and ≥ 10 years (n = 32), generating a mean L:EY of 0.29. The mean and standard deviation of the variance on these data (i.e., average and standard deviation of the individual variance values) was 0.024 and 0.015, respectively. Hence, there is considerable variability in the variance of L:EY across populations.
Most survivorship data we use originated from desert bighorn sheep within the Peninsular Ranges of California spanning 4-5 years 22 , with CV = 0.14. CVs of survivorships from these populations span 0.10 through 0.19, and CVs for the first 3 years of data average 14% lower than the CV from all years (SD = 0.27). Variation in survivorship also appears geographical and temporal.  Table 2. Values of population abundance for adult females (N), recruitment (R), and survivorship of adult females (S) used in the 30-year population modeling simulations (10,000 simulations per R and S combination) to acquire mean population growth (λ) ~ 1.0, with lower and upper 90% CI's (LCL, UCL), deterministic λ (λD), extinction probability at year 10 (E), quasi-extinction probability (QE; the probability that the population declines to ≤ 5 adult females at years 10 and 30), and the proportion of scenarios resulting in λ < 1.

Discussion
Given Caprinae life history and plausible combinations of mean recruitment and adult female survivorship, we evaluated population persistence and estimated population MVP. The values describing adult female survivorship and recruitment, plus the variability we employed match values found in other populations of Caprinae. We do not pool data across different Caprinae populations or species. Our approach and results directly inform the conservation and management of many Caprinae, especially those for which the acquisition of demographic data remains beyond reach. Our work embodies the characteristics of a high-quality PVA: clear objectives, appropriate demographic data, model structure matching species life histories, stochasticity, examination of extinction probability, appropriate time interval, use of mean values and associated variability 6 . As with most ecological models, the quest for more data remains problematic, not debilitating, and is addressed by creatively and aptly using existing information to generate meaningful results 3 .
Wildlife agencies generate lamb:adult female ratios from Caprinae surveys, recognizing that yearlings can be mistaken for adult females, causing miscounts. Excluding yearlings from the ratio's denominator assumes that no miscounts are occurring, yet an unknown and inconsistent number of yearlings remain in the adult female category across survey events. For these reasons, surveyors of other species, like Dall's sheep and caribou, pool counts of yearlings and adult females, generating lamb:"adult female-like" ratios instead 15,[23][24][25] .
Managers of Caprinae populations can follow these precedents and produce lamb:(adult female + yearling) ratios. Consistency would help standardize methods for building comparisons and meta-analyses across populations of Caprinae, while reducing variability across surveys due to differing techniques.
Typically, metrics like elasticity (proportional) and sensitivity (additive) describe the influences of demographic parameters on population growth 13,14,22,26 (30), the quasi-extinction value at year 30)). Mean stochastic lambda (λ) with 90% CIs (LCL, UCL) and probability of scenarios resulting in stochastic λ < 1 (deterministic λ = 1.03). Biological variation was derived from the empirical recruitment and survival data. www.nature.com/scientificreports/ general, the elasticity values for survival tend to be higher than those for recruitment 27 . Our results match this pattern, as the elasticity results indicate that a change in adult survival has a 2.5 times greater effect on λ than an equivalent change in recruitment. Relatedly, other theoretical work reports that demographic parameters with more temporal variability have lower elasticities, indicating less impact on population fitness (e.g. 28,29 ). Our work centers on applications. Since most management actions affect these demographic parameters simultaneously, at issue is the practicality (e.g. feasibility and affordability) of management to increase these parameters, and understanding how such changes could impact λ. For example, imagine a population with mean recruitment of 0.30 and adult survival 0.85, with a biologist interested in increasing recruitment or adult female survival to acquire λ ≥ 1. The answer is to increase either value by 0.02 (Fig. 1, Supplementary Data S1). Similarly, one can set a λ target and determine the amount of recruitment and adult female survival necessary for acquiring it (Fig. 1, Supplementary Data S1).

Minimum abundance target.
A minimum population of 50 adult females meets the persistence criteria, given intermediate levels of recruitment and survival producing λ ~ 1 ( Table 2). The risk of population collapse wanes as populations increase above the minimum threshold (Table 2; Fig. 1). For example, a population of ~ 100 adult females always meets persistence criteria ( Table 2). Populations of adult females should be somewhat larger than 50 when modest declines (λ ~ 0.97) are suspected, providing a cushion to address the causes of decline, and mitigate further reductions.
Translocation of 5 adult females during each of 5 years, or 10 in each of 3 years, requires a starting abundance of 70 adult females for the population to maintain the persistence criteria, never reach a lower confidence interval www.nature.com/scientificreports/ of 0, and for the population to return to the starting population size within 30 years. If managers mistakenly target a population having < 50 adult females, the population mean is unlikely to recover to pre-removal levels within 30 years (Fig. 4). The more adult females removed per year will reduce a populations abundance and elevate stochastic effects.

Applications. These survival and recruitment parameters have high temporal and geographical variability.
This uncertainty originates from the variability inherent to survivorship and recruitment data within Caprinae populations and demographic stochasticity operating over all age classes, making it difficult to predict exact values of λ and project abundances (Fig. 3). The causes of such variability are complex (e.g. factors such as variation in environmental drivers, predation, management types and timing, the distribution of female reproduction across populations and ages through time), and are rarely monitored. Without data, these factors cannot be explicitly examined. The effects of these factors, however, are contained in the empirically-derived estimates for survival and recruitment, and their associated variability. We examined the effects of time and management actions on reducing variation in recruitment and adult female survivorship. We began by halving the variance in recruitment, given its empirical basis. For all recruitment data, when the temporal window is short (3 years), recruitment variability declined to nearly half the overall mean variability. Also, the NMDGF operate a fenced facility (6. For survival, most of these data represented desert bighorn sheep populations whose abundance were declining (disease, predation, habitat loss 22 ). Survivorship of adult females in endangered or declining populations of ungulates could be lower or more variable than in demographically viable populations [30][31][32] .
Management actions can reduce variation in recruitment and survival. However, the simulations halving variance in recruitment or survival received minimal increases in precision in λ or abundance (Table 4; Fig. 3). Halving variability in adult female survivorship and recruitment, or eliminating it, moved the probability of λ < 1 to 0.39 and 0.31 (from 0.41) while reducing QE(30) from 0.05 to 0.01 and 0 respectively (Table 4). Therefore, in application, halving variance in recruitment (i.e., calculated from predator-free populations) and survival (i.e., from predator control) does not generate much change in population trajectories.
Management actions can also change the mean values of demographic parameters. For instance, populations of grazing mammals lacking predation have higher growth rates with less variability 33 . Therefore, when populations experience predation, predator control can improve population sizes of Caprinae by raising the survival of adult females and recruitment 12,34,35 . Survival of adult females can increase 5-12% with predator control, and survival with predation is often twice as variable as populations without predation 12 . Imagine predator control boosted mean adult female survival from 0.90 to 0.95 12 and recruitment from 0.30 to 0.50 (similar to the NM penned facility), while reducing the survival and recruitment variance by half ( 12 ; NM penned facility). Population growth would increase 20%, from λ = 1.02 to 1.22. Mean abundance would climb from 56 adult females to 135 by year 5.
Predators are important components of ecosystem health 36,37 , so predator management requires conscientious approaches. Predator control (or perhaps reduction in conspecific, alternative prey) could be applied to improve Caprinae populations with < 50 adult females. Larger populations should withstand background levels of predation, when stable or growing. Curbing declines in larger populations, or attaining targets in population growth or abundances could also warrant predator control, to accomplish management objectives. Monitoring the abundance and recruitment of Caprinae populations helps identify conditions suggesting management actions like predator control, while indicating when the desired project objectives are achieved (i.e. Caprinae www.nature.com/scientificreports/ population responses). Factors like environment, climate and disease influence population abundance too, and are worth considering during such monitoring 38 . Provision of supplemental water can increase the distributions of Caprinae 39,40 . Supplemental water, however, could increase predator distributions (directly or indirectly), potentially aggravating predation issues affecting adult female survival and recruitment 41,42 . Other hypotheses, like temporal reductions in water supplementation, may also reduce predator presence within Caprinae ranges 43,44 . Alternative management tools for improving Caprinae abundances are supplemental feeding and disease control, challenging and costly to pursue at large scales.
The variability in adult female survival and recruitment means that 1 year's recruitment or survival cannot accurately predict the next, and historical trends in population abundance may have little bearing on future performance (Fig. 3). Revealing management contributions to population growth is also challenged by the variability inherent within the demographics of Caprinae populations. This situation makes Caprinae management reactive, with management decisions based on the short-term monitoring results for a population. The repeated acquisition of accurate abundance data, however, builds a more proactive paradigm.
Monitoring Caprinae. Population vital rates and their importance to population growth often varies across populations and within a population over time (our results 31,32,45,46 ). Indeed, within and across populations of Caprinae, the annual differences in survival and recruitment are substantial 47,48 . Some temporal and geographical variation is biological, and some stems from the use of different methodologies for quantifying variability. Hence, analytical projections of future abundance based on survival and recruitment data from one population are unlikely to apply to other populations for the same or different species. Understanding a given populations status requires directly monitoring that population.
Frequent monitoring of many populations requires simple and affordable methods for identifying a population's status (abundance, growth). We discovered the threshold of 50 adult females with three types of demographic parameters: abundance, survival and recruitment. Monitoring the threshold is achievable with two of them.
Adult female survival is the most challenging and costly parameter to acquire, unattainable for most biologists managing Caprinae populations. Indeed, we struggled to locate data describing adult female survival given a species extensively studied. Future monitoring and analyses, therefore, do not require survival data, but can rely solely on measures of abundance and recruitment.
Minimum counts are the primary method for monitoring Caprinae species. This method is appropriate if precision measurements are not required and acknowledgement that data are biased low. With minimum counts, management actions dependent upon abundance triggers (i.e., 50 adult females) would be identified early, providing a conservative approach to population management. In the short term, populations can be prioritized for conservation using minimum abundances, but these data make it difficult to generate longer-term conclusions about population sustainability.
When minimum counts or mean abundance estimates for adult females are well above 50 (i.e. ~ ≥ 75), the population is likely to persist (unless conditions change). When populations occur at or near the threshold value, recruitment estimates associated with these abundance estimates (and trends) help biologists infer if the population is stable or growing, and if the minimum abundance threshold were reached. For example, imagine a population with known abundance of 70 adult females. A survey was conducted, producing an estimate of 65 adult females (90% CI 44-86 (20% CV)), with a recruitment estimate (Lamb:(adult female + yearling)) > 0.30. Trends in abundance from prior surveys indicate an increase. A reasonable interpretation of these data is that the population is likely at or above the threshold and growing.
We used abundance trends because of the high variability in recruitment trends (demonstrated herein). If either abundance, abundance trend or recruitment shows a concerning decline, it should initiate work to discern the cause, as the population may no longer be sustainable.
While the acquisition of accurate abundance data can be challenging [48][49][50] , researchers have posed tenable solutions for meeting these challenges 50,51 . These methods rely on methodological changes in aerial surveys, or replacing aerial surveys with motion activated cameras, that collaterally identify predators and their relative abundances.
For annually-reproducing Caprinae that begin reproduction in their second to third year, the minimum abundance threshold of 50 adult females should apply. Source populations for translocation have a 70 adult female abundance threshold, assuming translocations of 5 or 10 adult females per year, in each of 5 and 3 years respectively. For Caprinae with different life histories, our model code is easily modified to accommodate those changes.
Hence, our work extends beyond Caprinae. The population projection matrix model we developed can be applied to any species reproducing on an annual cycle. Users specify parameters such as the number of stageclasses, mean and variance of recruitment and adult survival, starting population size, correlation between recruitment and survival, sex-ratio at recruitment, and number of translocations (Supplementary Data S1). As exemplified herein, results will identify a population's'minimum threshold for persistence, which subsequently informs threat assessments, harvest quotas and the triage of conservation activities aimed at recovering ailing populations.

Methods
Our matrix model simulated population trajectories given various combinations of starting abundances, recruitment, adult female survival values, and translocation scenarios (Fig. 5). We predicted population growth, adult female abundance and determined extinction and quasi-extinction (QE) probabilities through time. We defined extinction when abundance = 0 adult females, and QE when abundance ≤ 5 adult females. Results from these We found few publications describing ewe survival for Caprinae. Therefore, to compute variability, we relied on survival data for desert bighorn sheep from three publications with studies spanning multiple, sequential years 22,52,53  Desert bighorn sheep share these characteristics. We estimated the biological variance (i.e., the component of variance due to temporal and environmental variation) of survival and recruitment by using the variance discounting procedures which removes the sampling variation 54 .
Population dynamics. The stage-structured population projection matrix model used 11 age-based categories (Fig. 5). Our models used age-class delineations corresponding to a post-breeding census. We considered lambs (or kids) < 0.5 years, yearlings as ages ≥ 0.5 and < 1.5 years, and adult females as ≥ 1.5 and ≤ 12.5 years (herein, the term "lamb" is also interchangeable for "kid", when the sentence is used in context for Caprinae species). We set yearling survivorship as 10% lower than adult female survivorship, a reduction based on prior work with bighorn sheep 55 . We considered prime aged animals ages 2-10 years, with survival equivalent across these age classes 55,56 . Older females (≥ 11.5) received a 0.43 proportional reduction in survival from adult female survival (calculated from life history table 56 ). We set the amount of variation in yearling and older female survival to equal the variation in adult survival.
We integrated demographic stochasticity (inherent randomness that arises from the discrete nature of individual animals 57 ) and biological variation into the simulation. Biological variation was derived from the empirical recruitment and survival data. For each year and iteration, we randomly selected adult female survival and lamb recruitment using the incomplete beta function, which allows for correlated vital rates 58 . We assumed the vital rates were correlated at 0.40 (calculated from 22 ). We used the first and second moments to calculate the scale and shape parameters of the beta distribution 58,59 . We assumed a 50 male:50 female sex ratio of recruited lambs, normally distributed with a variance of 0.01 60 . We incorporated demographic stochasticity as a binomial random variable through the random generation of survivors and recruits. The number of trials (n) matched the number of individual adult females in a given stage, with the parameter p of the binomial distribution a random variable simulated from the beta distribution.
Our simulations used combinations of mean survival and recruitment values to estimate population growth (λ). We allowed mean survival to range from 0.70 to 0.98, mean recruitment from 0.05 to 0.90 (to account for scenarios with and without twinning) and starting abundance from 10 to 250 adult females. Biological variance www.nature.com/scientificreports/ remained fixed to empirical values. We explored the effects of biological variation on population dynamics in subsequent simulations by artificially halving and eliminating the recruitment and survivorship variances. We also identified how changes in survivorship and recruitment affected λ. While multiple factors affect the mortality of adult females (e.g. predation, starvation, disease), declines in abundance also occurs when adult females are translocated from a source population to another population. Translocations of adult females should be an acceptable practice, provided the source population does not encumber a high risk of QE resulting from removals. When desert bighorn sheep are translocated from a population, it is common for managers to pursue sequential years of removals. Therefore, we examined the effects of adult female removals on population stability by examining two translocation scenarios: removing 5 adult females consecutively for 5 years (25 total removals) and removing 10 adult females over 3 consecutive years (30 total removals). These translocation simulations fixed recruitment and survival parameters to generate a stochastic λ = 1.02 (recruitment = 0.30, survivorship = 0.90, with biological variability), as populations with λ ≤ 1.00 would not recover from the translocation event.
Simulations estimated abundance and λ for each combination of survival and recruitment, every year, for 30 years, with 10,000 individual iterations. We tested model operation to ensure results replicated the values of mean recruitment, mean survival and their respective variability. For each simulation we determined extinction and QE each year.
Data describing lamb:adult female ratios typically represent recruitment, a combination of adult female fecundity and lamb survival. Lambs are born in spring and surviving lambs are counted and considered recruited during fall surveys (~ 6 months post-birthing). Our model relies on recruitment ratios, as these data are routinely collected by Caprinae biologists. For Caprinae, age-specific recruitment data are rarely (if ever) recorded during population monitoring. The variability in the lamb:ewe data accounts for any differences in reproduction across age classes through time and across populations.
Management agencies report lamb:adult female ratios differently, with some agencies including or excluding yearlings in the ratio denominator. We examined these lamb:adult female data, finding that yearlings (of both sexes) were often misclassified as adult females. This outcome is not unusual, as surveys for other species often misclassify yearling male and females as adult females 25 . Grouping yearlings and adult females into a ratio of lamb:'adult female-like' is also a common solution, for representing recruitment in Caprinae and other ungulates 15,24,61 . Therefore, we employed lamb:'adult female-like' ratios in the simulations.
Mathematically, we adjusted the population projection matrix to account for the low bias in recruitment, occurring from the inclusion of yearling males and females in the denominator of the ratio. This simple adjustment multiplied the recruitment ratio by 2 in the first column of the matrix (bold R representing the first age category; Fig. 5; Supplementary Data S1). To explain, imagine a hypothetical population consisting of 80 adult females (E) producing 30 lambs (L), with 10 male and 10 female yearlings (YM and YF respectively). The conventional lamb:adult female ratio (L:E) is 0.375. An unknown number of yearlings, however, are misclassified as adult females. When all yearlings are included in the denominator of the lamb:adult female ratio (L:(E + YM + YF)) the recruitment ratio is biased low (0.30). The traditional matrix would only include R in the 2nd-11th column of the matrix (the bold R omitted in the matrix within Fig. 5), which would produce 24 lambs (80 × 0.30 = 24). Yet we know from the example, that 30 lambs were produced. Alternatively, if R occurred in all columns of the matrix, the result is 27 lambs (80 × 0.30 + 10 × 0.30 = 27) and lamb production remains biased low. This problem arises because male and female yearlings are not reliably distinguished. To correct this problem, we multiplied R by 2 in the first column (80 × 0.30 + 10 × 2 × 0.30 = 30), which accounts for yearling males. The 30 lambs includes both sexes, and since our simulation represents adult females, R is multiplied by the lamb sex ratio (c = 0.5).